Matrices and Determinants

Abstract

In the second chapter we deal with matrices and determinants. The chapter starts with determinants of second and third orders, which are defined through solutions of linear algebraic systems; determinants of arbitrary order are defined inductively. The basic properties of determinants are investigated. We then take a look at determinants from a more abstract viewpoint: it is proved that the determinant of a square matrix can be defined as an antisymmetric multilinear function of the rows. Using some basic elements of permutation theory, we continue to study the properties of determinants; in particular, we derive explicit formula for determinants. Finally, we define the rank of a matrix and the main operations on matrices (sum, product, inverse matrix) and investigate their properties.